A Modified Mean Stress Criterion for Considering Size Effects on Mode I Fracture Estimation of Rounded-Tip V-Notched Polymeric Specimens

The aim of this paper is to assess the size and geometry effects on the mode I notch fracture toughness of polymeric samples containing rounded-tip V-shaped (RV) notches (V-notch with a finite radius at the notch tip). First, using a large number of fracture tests on an RV-notched Brazilian disk and semi-circular bending polymeric samples with four different sizes, the size-dependent values of the notch fracture toughness are obtained. Then, the mean stress criterion is modified for characterizing the size-dependency of notch fracture toughness in polymeric samples. The resulting modified mean stress criterion considers higher order terms of the stress field when calculating the fracture process zone length around the tip of the defect. Additionally, the critical distance rc is assumed to be associated with the specimen size and a formula containing fitting parameters is utilized for considering this trend of rc. The comparison between the values of notch fracture toughness obtained from experiments and those predicted by the modified mean stress criterion shows that the suggested approach can provide accurate estimations of size-dependent values of notch fracture toughness in polymeric specimens containing RV notches.


Introduction
Polymeric materials such as polymethyl-methacrylate (PMMA) and general-purpose polystyrene (GPPS) are widely used in industrial applications due to their good physical and mechanical properties. PMMA (or plexiglass ® ) is an artificial rigid amorphous polymer that can be combined with laser cutting, forming or bending processes. GPPS (also known as crystal polystyrene) is a thermoplastic multi-purpose polymer with brittle behavior and excellent X-ray resistance, low contraction and low production cost. Similar to other polymers, the mechanical properties of both PMMA and GPPS can be improved by adding a suitable percentage of short or long fibers [1][2][3]. Fibers and their orientations affect and control the damage mechanisms of polymers including fiber pull-out, growth of hackles, profuse crazing, energy absorbing, etc. [4,5].
Due to design, lubrication and/or optimization requirements, notch-type defects such as round tip V-shaped (RV) notches may be present in polymeric components. These notches lead to stress concentrations at their vicinity and the load bearing capacity of the corresponding notched components decreases. Therefore, the fracture assessment of notched parts has always been of interest for structural engineers and researchers, particularly when dealing with brittle and quasi-brittle materials such as polymers, given that the fracture process of these materials is generally sudden and catastrophic. radii were prepared and tested. It is shown that the values of the NFT obtained from fracture tests on RVNBD and RVSCB specimens are size-dependent. In order to explain the size dependency of the NFT, a variant of the mean stress criterion, namely MS-Schmidt, used frequently for fracture analyses of notched samples is applied. Additionally, for the first time, the modified mean stress criterion (MMS) is introduced, which considers higher order terms of the stress field when calculating the FPZ length around the defect tip. In both the MS-Schmidt and the MMS criteria, the length of the FPZ is considered to be dependent on the size of the specimen being analyzed. The size-dependent values of the FPZ length are also defined for cracked Brazilian disk (CBD) and cracked semi-circular bending (SCB) specimens with various radii employing a semi-empirical relation. It is finally observed how the suggested criteria successfully predict the mode I NFR of PMMA and GPPS materials considering size and geometry effects.

Material Identification
The materials analyzed in this work are PMMA and GPPS, manufactured by the Cho Chen Ind. Company, Tainan, Taiwan. Following the specifications of the material suppliers, the average ultimate tensile strength (σ u ), Young's modulus (E) and Poisson's ratio (υ) for the PMMA are, respectively, 75.8 MPa, 3.45 GPa and 0.38, while the values of these mechanical properties for GPPS are 45.0 MPa, 3.1 GPa and 0.34, respectively. The tensile strength of PMMA and GPPS is a very important parameter when calculating the FPZ length. Therefore, five dog-bone samples were prepared according to ASTM D638-14 [49] and EN ISO 527 [50] standards for each material and then tested under unidirectional tensile load to acquire a reliable value of σ u . These unidirectional tensile tests indicated that the values of 75.8 and 45 MPa reported by supplier companies were accurate, and, thus, these values will be considered in this research. Moreover, the PMMA and GPPS sheets provided in this study were produced by the extrusion technique and their mechanical properties might be different in various directions. Therefore, the isotropy of both PMMA and GPPS materials were demonstrated by testing the tensile samples in two perpendicular directions.

Test Configuration and Preparation of Fracture Specimens
In order to assess the size effect on the NFR of PMMA and GPPS materials, RV-notched Brazilian disk (RVNBD) and RV-notched semi-circular bend (RVSCB) specimens were employed. Figure 1 presents the schematics of RVNBD and RVSCB specimens. RVNBD is a circular disk with radius R = D/2 (D is the diameter of BD sample) and thickness t, having a central rhombic hole in which the large diagonal and notch opening angle are, respectively d = 2a and 2α (a being half of the defect length). The lower and upper corners of the rhombic hole are blunted by the radius of ρ, generating a RV-notch. The RVSCB is a semi-circular disk with a radius of R and thickness of t, in which a RV-notch is generated on the edge of the disk. The angle and length of the resulting RV-notch are 2α and a, respectively. The loading condition in the RVSCB sample is provided by a three-point fixture in which the spam between the supports is 2S. When the direction of the applied load and the bisector line of the RV notch are the same in the RVSCB specimen, and the bottom supports are symmetric relative to this direction, this sample is subjected to pure mode I. Likewise, pure mode I loading is archived in the RVNBD sample by setting the applied load along the bisector line of the RV notch.
Since the fracture toughness K Ic of PMMA and GPPS materials must be specified for calculating the FPZ length, several cracked Brazilian disk (CBD) and semi-circular bend (SCB) specimens were prepared and tested. The CBD and SCB specimens are similar to the RVNBD and RVSCB samples, except that a sharp crack of length 2a for CBD and length a for SCB is generated in the center of the sample instead of in the rhombic hole and other configurations remain constant. Table 1 gathers the dimensions and loading conditions for all samples. Since the fracture toughness KIc of PMMA and GPPS materials must be specifie calculating the FPZ length, several cracked Brazilian disk (CBD) and semi-circular (SCB) specimens were prepared and tested. The CBD and SCB specimens are simi the RVNBD and RVSCB samples, except that a sharp crack of length 2a for CBD and le a for SCB is generated in the center of the sample instead of in the rhombic hole and configurations remain constant. Table 1 gathers the dimensions and loading condi for all samples.  CBD   5  5  --10  10  --20  20  --35  35  --SCB   5  2.5  8  -10  5  16  -20  10  32  -35 17 . All samples and notches were cut by a waterjet machine from PMMA and GP mm thick sheets. The cracks in both CBD and SCB were created by jigsaw with a thick of 0.2 mm. Then, the generated cracks were sharpened by way of a razor blade w thickness of 40 µm. Figure 2 shows a schematic of all tested specimens with various  All samples and notches were cut by a waterjet machine from PMMA and GPPS 6 mm thick sheets. The cracks in both CBD and SCB were created by jigsaw with a thickness of 0.2 mm. Then, the generated cracks were sharpened by way of a razor blade with a thickness of 40 µm. Figure 2 shows a schematic of all tested specimens with various sizes.

Fracture Tests
All notched and cracked samples were tested by using a universal test machine. The tests were carried out by displacement control conditions. The speed of the crosshead was fixed at 0.5 mm/min for all tests, which were continued until final fracture and separation of the two halves of the samples. An LS-20 load cell (2000 kgf), with a maximum error of 0.5% in the range of 2-100% of the nominal capacity was utilized. A load cell with a capacity of 500 kgf was also employed to enhance the accuracy of measurements for smaller specimens with nominal radii R5 and R10 mm.
. Figure 2. Schematic of scaling on the nominal radius of BD and SCB specimens containing crack and RV notch.

Fracture Tests
All notched and cracked samples were tested by using a universal test machine. The tests were carried out by displacement control conditions. The speed of the crosshead was fixed at 0.5 mm/min for all tests, which were continued until final fracture and separation of the two halves of the samples. An LS-20 load cell (2000 kgf), with a maximum error of 0.5% in the range of 2-100% of the nominal capacity was utilized. A load cell with a capacity of 500 kgf was also employed to enhance the accuracy of measurements for smaller specimens with nominal radii R5 and R10 mm.
The load-displacement curve for each test was recorded, as shown in the example of Figure 3. The load-displacement curves were nearly linear until final fracture, except at the initial moment when a slip occurred between the upper fixture roller and the BD and SCB samples, with no effect on the fracture load. It is noteworthy that experiments on the The load-displacement curve for each test was recorded, as shown in the example of Figure 3. The load-displacement curves were nearly linear until final fracture, except at the initial moment when a slip occurred between the upper fixture roller and the BD and SCB samples, with no effect on the fracture load. It is noteworthy that experiments on the SCB samples were conducted with two different fixtures due to the different nominal dimensions of SCB specimens. In other words, the two largest specimens (i.e., samples with R = 20 and 35 mm) were loaded using one fixture, while the two smallest ones (i.e., samples with R = 5 and 10 mm) were tested using another fixture with sharper edges. Tables 2 and 3 show the fracture loads P f in all the tested specimens (cracked and notched, respectively). SCB samples were conducted with two different fixtures due to the different nominal dimensions of SCB specimens. In other words, the two largest specimens (i.e., samples with R = 20 and 35 mm) were loaded using one fixture, while the two smallest ones (i.e., samples with R = 5 and 10 mm) were tested using another fixture with sharper edges. Tables  2 and 3 show the fracture loads Pf in all the tested specimens (cracked and notched, respectively).    In addition, Figure 4 shows an example of fractured RVNBD and RVSCB specimens of GPPS and PMMA materials. It can be observed how cracks propagated along the bisector line of the initial notch in RVNBD and RVSCB samples. In addition, Figure 4 shows an example of fractured RVNBD and RVSCB specimens of GPPS and PMMA materials. It can be observed how cracks propagated along the bisector line of the initial notch in RVNBD and RVSCB samples.

Fracture Toughness and Notch Fracture Toughness Values
The experimental values of the fracture toughness (KIc) of GPPS and PMMA materials may be derived from the CBD and the SCB (cracked) samples. In the case of the SCB samples, KIc values will be obtained by using the relationship proposed by ISRM standard [51]: where S is the half distance between bottom supports, R is the radius of the SCB sample, t is the sample thickness, a is the crack length, Pmax is the fracture load and YI is the geometrical factor, which follows Equation (2) Concerning the CBD samples, the equation suggested by Akbardoost and Ayatollahi [52] will be used for determining the value of KIc: where R and t are, respectively, the radius and the thickness of the CBD sample, Pmax is the fracture load, and KI * is the dimensionless geometry factor, equal to 0.2208 when a/R = 0.5 [52].

Fracture Toughness and Notch Fracture Toughness Values
The experimental values of the fracture toughness (K Ic ) of GPPS and PMMA materials may be derived from the CBD and the SCB (cracked) samples. In the case of the SCB samples, K Ic values will be obtained by using the relationship proposed by ISRM standard [51]: where S is the half distance between bottom supports, R is the radius of the SCB sample, t is the sample thickness, a is the crack length, P max is the fracture load and Y I is the geometrical factor, which follows Equation (2) Concerning the CBD samples, the equation suggested by Akbardoost and Ayatollahi [52] will be used for determining the value of K Ic : where R and t are, respectively, the radius and the thickness of the CBD sample, P max is the fracture load, and K I * is the dimensionless geometry factor, equal to 0.2208 when a/R = 0.5 [52].
Similarly, notch fracture toughness (K Ic V,ρ ) values may be derived from the tests performed on notched specimens. Lazzarin and Filippi [53] proposed the following equation to determine the notch stress intensity factor: where r 0 is the distance from the origin of the coordinate system to the notch tip [53], λ 1 is a William's mode I eigenvalue [53], ω 1 is an auxiliary parameter depending on the opening angle (α) [53,54], and σ θθ (r 0 ,0) is the tangential stress at the notch tip, usually obtained from finite element (FE) analyses. The calculation of the notch fracture toughness (K Ic V,ρ ) requires, firstly, to develop a (linear-elastic) FE analysis of the corresponding sample and to subject the virtual specimen to the averaged fracture load measured in the experiments, obtaining the tangential stress component along the notch bisector. Then, the value of σ θθ (r 0 ,0) calculated from FE analysis is substituted into Equation (4). The details of the finite element analyses performed in this research to determine the different values of notch fracture toughness are summarized in Appendix A.

MS-Schmidt and MMS Criteria for the Analysis of the Size Effect
Two stress-based brittle fracture criteria for considering the size effect are evaluated in this section. These criteria are the mean stress criterion as defined by Schmidt [55] (MS-Schmidt) and the modified mean stress (MMS) criterion. The basis in both cases is the same as that established by the classical mean stress (MS) criterion: the onset of fracture in the notched samples is achieved when the tangential stress averaged over a critical distance (d c ) attains a critical value of (σ θθ ) c . For brittle materials, the value of (σ θθ ) c in the MS criterion is assumed to be the material tensile strength σ u [56,57].
Following and simplifying the MS criterion, Ayatollahi and Torabi [6,8] stated that mode I fracture in notched conditions occurs when the notch stress intensity factor (NSIF) K I V,ρ reaches the notch fracture toughness K Ic V,ρ . They also proposed a relation to predict the value of K Ic V,ρ : According to [8]: The eigenvalues λ 1 and µ 1 , as well as the values of χ b1 , χ c1 and χ d1 , depend on the notch opening angle (2α) [54].
The critical distance d c,V is an important parameter in Equation (5). RV notches follow Equation (7): where, again, r 0 is the distance between the origin of the coordinate axes and the notch tip. According to [54], r 0 can be acquired as follows: d c in Equation (7) is the theoretical length of the fracture process zone (FPZ), based on the MS criterion, and can be obtained from Schmidt's formula [55] for cracked conditions: d c-Schmidt = (2/π)·(K Ic /σ u ) 2 (9) Figure 5 shows a schematic of d c,V in RV notches. The MS criterion, in which the value of d c is obtained from Schmidt's formula, will be denoted hereafter as the MS-Schmidt criterion. A second variant of the MS criterion for considering the size effect is the MMS criterion, which considers the higher order terms of the Williams series solution of the stress field when calculating the value of dc in a cracked specimen. More precisely, the tangential stress component is defined by taking into account the first three terms of the Williams A second variant of the MS criterion for considering the size effect is the MMS criterion, which considers the higher order terms of the Williams series solution of the stress field when calculating the value of d c in a cracked specimen. More precisely, the tangential stress component is defined by taking into account the first three terms of the Williams series expansion in pure mode I loading [27]: where the parameters A 1 * and A 3 * are the dimensionless coefficients of the first and third terms of the Williams series expansion. Then, Equation (11) is used for calculating the tangential stress averaged over the critical distance d c : (1/d c )· 0 dc σ θθ· dr = (σ θθ ) c = σ u , The final equation for d c when using the MMS gives: d c-MMS = (π/8)·{[R·A 1 *·σ u ∓ ((R·A 1 *·σ u ) 2 − (8·A 1 *·A 3 *·R·K Ic 2 /π)) 1/2 ]/(A 3 *·K Ic )} 2 , (12) In Equation (12), the minimum positive value of d c is physically acceptable and will be denoted as d c-MMS for the sake of more clarity. Now, d c,V can be calculated according to the new proposed approach as follows: The length of the FPZ depends on the size of the cracked specimen [15,24,27]. Therefore, it can be stated that the parameter d c in both the approaches mentioned above (i.e., d c-MMS and d c-Schmidt ) is size-dependent. In the present study, the recent approach suggested by Ayatollahi and Akbardoost [27] is used to explore the size effect on the parameter d c : in which A and B are fitting parameters that are determined from mode I fracture tests performed in cracked samples with different sizes. Equation (14) may be converted into Equation (15), which is more suitable for linear regression: More details about the calculation of A and B will be described below. Now, the size-dependent values of the notch fracture toughness for notched polymeric samples can be predicted by replacing the parameter d c according to the size of the sample from Equations (9) or (12) into Equations (7) or (13) and then substituting the calculated values of d c,V into Equation (5). In the next sections, the notch fracture toughness obtained from experiments will be compared with the predictions derived from the MS criteria (MS-Schmidt's and MMS).

Results and Discussion
The fracture toughness (K Ic ) of GPPS and PMMA materials are easily derived by substituting the fracture loads and sample dimensions listed in Tables 1 and 2 into Equations (1) and (3), using P f (average values of fracture loads) as P max . Table 4 gathers the resulting values of K Ic , where it can be observed that the results are dependent on the size of the specimen, with larger values in the larger samples.  Table 5 gathers the values of the notch fracture toughness obtained for PMMA and GPPS in RVNBD and RVSCB specimens. As in the K Ic values, Table 5 reveals that K Ic V,ρ depends on the size of the specimen and becomes larger as the size of the specimen grows.  The determination of d c for each combination of sample size, sample type (CBD and SCB specimens) and material (PMMA and GPPS) is the first step in the MS-Schmidt's and MMS criteria. To do this, the values of K Ic values listed in Table 4 are replaced into Equations (9) and (12). Additionally, the tensile strength σ u in these two equations is considered as 75.8 MPa for PMMA and 45.0 MPa for GPPS. The results are shown in Table 6, where it can be observed again that d c changes by altering the nominal dimension of the sample. To quantify the size-dependency of d c , Equation (14) is utilized, while the fitting parameters A and B are determined from a linear regression between the variations of 1/d c versus 1/R, as shown in Figure 6. Once the parameters A and B are calculated for each material and sample, the evolution of d c with the sample size can be found.  Now, the size-dependent values of dc from Equation (15) for each material are replaced into Equations (7) and (13) in order to determine the corresponding value of dc,V for each sample size. After that, the mode I notch fracture toughness value (KIc V,ρ ) for each tested specimen is predicted by substituting the parameter dc,V into Equation (5). The variations of KIc V,ρ with the nominal radius (i.e., size) of the specimens are predicted by the MS-Schmidt's and the MMS criteria and compared with those obtained experimentally from the RVNBD and the RVSCB samples made of PMMA and GPPS, with the results being shown in Figures 7-10. According to these figures, it is observable that the MS-Schmidt's and the MMS criteria provide good estimations of KIc V,ρ in BD and SCB samples made of PMMA and GPPS polymers containing RV notches.  (7) and (13) in order to determine the corresponding value of d c,V for each sample size. After that, the mode I notch fracture toughness value (K Ic V,ρ ) for each tested specimen is predicted by substituting the parameter d c,V into Equation (5). The variations of K Ic V,ρ with the nominal radius (i.e., size) of the specimens are predicted by the MS-Schmidt's and the MMS criteria and compared with those obtained experimentally from the RVNBD and the RVSCB samples made of PMMA and GPPS, with the results being shown in Figures 7-10. According to these figures, it is observable that the MS-Schmidt's and the MMS criteria provide good estimations of K Ic V,ρ in BD and SCB samples made of PMMA and GPPS polymers containing RV notches.
The results shown in Figures 7-10 reveal slight deviations between the experimental data and the theoretical predictions, with such (very moderate) deviations being more significant in RVNSCB specimens. Predictions are accurate for the whole range of sample sizes analyzed in this work and for the two criteria applied in this research.
In addition, the two criteria provide, in practical terms, the same predictions in RVNBD specimens, and very similar predictions in RVNSCB specimens. Since Equation (9) is simpler than Equation (12) for calculating the parameter d c , this formula may be preferred for the design or analysis of notched structures manufactured from PMMA or GPPS materials. Significantly larger differences between the fracture loads predicted by the MS-Schmidt and the MMS criteria can be expected for those engineering materials having generally large FPZs, such as concrete, rock, etc. tested specimen is predicted by substituting the parameter dc,V into Equation (5). The variations of KIc V,ρ with the nominal radius (i.e., size) of the specimens are predicted by the MS-Schmidt's and the MMS criteria and compared with those obtained experimentally from the RVNBD and the RVSCB samples made of PMMA and GPPS, with the results being shown in Figures 7-10. According to these figures, it is observable that the MS-Schmidt's and the MMS criteria provide good estimations of KIc V,ρ in BD and SCB samples made of PMMA and GPPS polymers containing RV notches.   The results shown in Figures 7-10 reveal slight deviations between the experimental data and the theoretical predictions, with such (very moderate) deviations being more significant in RVNSCB specimens. Predictions are accurate for the whole range of sample sizes analyzed in this work and for the two criteria applied in this research.
In addition, the two criteria provide, in practical terms, the same predictions in RVNBD specimens, and very similar predictions in RVNSCB specimens. Since Equation (9) is simpler than Equation (12) for calculating the parameter dc, this formula may be preferred for the design or analysis of notched structures manufactured from PMMA or

Conclusions
In this study, it was observed that the values of fracture toughness and notch fracture toughness (K Ic V,ρ ) of PMMA and GPPS clearly depend on the size of the samples being tested, in such a way that the fracture resistance of polymeric cracked and notched samples enhances by increasing their nominal size.
In this way, a large number of RV-notched Brazilian disk (RVNBD) and RV-notched semi-circular bending (RVNSCB) specimens containing different notch radii were prepared from PMMA and GPPS sheets and then tested in pure mode I loading.
Two alternative versions of the mean stress (MS) criterion were developed for justifying the size effect on notch fracture resistance K Ic V,ρ , referred to as the MS-Schmidt criterion and the modified mean stress (MMS) criterion. Thus, two formulations were considered for determining the critical distance d c : the first one is the relation proposed by Schmidt and the second one is the formula in which the higher order terms in the Williams series expansion are considered. In both cases, the critical distance depends on the size of the specimen.
The small discrepancies between the predictions of notch fracture toughness provided both modifications of the MS criterion and the experimental results, observed in all RVNBD and RVNSCB polymeric specimens, demonstrate the ability of both criteria to predict the mode I notch fracture toughness at different scales. It can be stated that in polymeric materials such as PMMA and GPPS, which have a small fracture process zone (FPZ) length, the MS-Schmidt criterion is more efficient due to its combination of accuracy and simplicity.

Data Availability Statement:
The data presented in this study are available on request from the corresponding authors.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Finite element analyses were performed in pure mode I for determining the critical stress intensity factors (SIFs) and critical notch stress intensity factors (NSIFs) for the cracked and notched polymeric BD and SCB specimens. Specimens were subjected to the corresponding fracture loads. All RVNBD and RVNSCB samples were modeled in two-dimensional form in Abaqus 6.14 code and were meshed by using CPS8R-type elements, which are quadrilateral, 8-nodes, biquadratic, plane stress and reduced integration elements.
It should be noted that the elements around the notch border were very fine with the aim of considering the high stress gradient at the vicinity of these areas. The size of the element for each sample is dependent on its size. For example, the mesh size for samples with a nominal radius of R5 is considered to be 0.01 mm around the notch tip, while its size is 0.1 mm for far-field. For larger specimens, the size of element increases proportionally in both near-field and far-field zones. Figure A1 shows the partitioning of RVNBD and RVNSCB samples, as well as their boundary and loading conditions. An example of the mesh pattern is also represented in Figure A2 for both specimens. After numerical computations in ABAQUS, the tangential stress component at the notch tip σ θθ (r 0 ,0) is obtained. Figure A3 displays a typical contour of tangential stress for RVNBD and RVNSCB specimens. The values of σ θθ (r 0 ,0) extracted from FEA are substituted into Equation (4) for calculating K I V,ρ .
ally in both near-field and far-field zones. Figure A1 shows the partitioning of RVNBD and RVNSCB samples, as well as their boundary and loading conditions. An example of the mesh pattern is also represented in Figure A2 for both specimens. After numerical computations in ABAQUS, the tangential stress component at the notch tip σθθ(r0,0) is obtained. Figure A3 displays a typical contour of tangential stress for RVNBD and RVNSCB specimens. The values of σθθ(r0,0) extracted from FEA are substituted into Equation (4) for calculating KI V,ρ . Figure A1. The boundary conditions and partitioning for RVNBD and RVNSCB samples. Figure A1. The boundary conditions and partitioning for RVNBD and RVNSCB samples.